Abstract: I will introduce the concept of Krylov complexity, which quantifies the spread of an initial state or operator under Hamiltonian time evolution. It has been very successful in probing, especially quantum chaotic dynamics, in a broad range of quantum systems, starting from spin systems to holographic setups. After introducing the basic methodology and some of its important success stories, I will briefly discuss some of my recent works in this direction on highly excited string scattering (HESS) and measurement-induced random quantum circuits. In different ways, both of these models are considered to be setups which can provide us with insights about information scrambling inside black holes. While the complexity analysis of HESS puts the conjecture of black holes being maximally chaotic objects on a stronger footing, the random circuit analysis brings the idea of discrete time Krylov complexity to the broadly known concept of circuit complexity, which quantifies the optimal number of quantum gates needed to construct a target state starting from a given initial state.